Banks struck with a rising effort in the 15th minute that beat Martin Rice from 18 yards even though the goalkeeper got a big hand to the strike.
Chesterfield surged further ahead four minutes later as Jay O'Shea beat Rice, who initially made a superb point-blank save, from close-range on the follow-up.
Rice denied O'Shea with another superb save at the start of the second half as Chesterfield again took the game to their struggling hosts.
Cook said: "I think we gave a really professional performance and saw the game out properly.
"It was the way we should have possibly seen the game out at Morecambe and possibly other places.
"We had a lot of criticism for that but the players were excellent; our senior players at the back, our seniors through the middle of the pitch and our senior players upfront players like Gary Roberts, Sammy Hird and Ian Evatt, were absolutely immense.
"That's what we ask for and that was the disappointment of Morecambe, but that's well gone now.
"Oli Banks has come in and looks like he has been playing for us for years, he gave everyone a great lift at a great time.
"Possibly if I am being critical we could have been a bit more forceful with our play to score one or two more but sometimes you are never happy as a manager and we are going back up the road and I am very pleased.
"The lads were excellent at the end of a tough week. We asked for three wins at the club. You know it's very hard to win two games on the bounce in any football but to win three on the bounce is great credit to the players and they should take all the plaudits."
- Sports & Recreation
- American Football
- Martin Rice